The properties of the material undergoing thermal processing play a very important role in the mathematical and numerical modeling of the process.
The variation of dynamic viscosity µ requires special consideration for materials such as lubricants, plastics, polymers, food materials and several oils, that are of interest in a variety of manufacturing processes. Most of these materials are non-Newtonian in behavior, implying that the shear stress is not proportional to the shear rate. The viscosityµis a function of the shear rate. For Newtonian fluids like air and water, the viscosity is independent of the shear rate, but increases or decreases with the shear rate for shear thickening or thinning fluids, respectively. These are viscoinelastic fluids, which may be time-independent or time-dependent.
The mechanical behavior of a material and its mechanical or rheological properties can be characterized in terms how the shear stress and shear rate are related. If the properties of the fluid are such that the shear stress and shear rate are proportional, the material is known as aNewtonian fluid. This relationship in the boundary layer is expressed by
(1.8) τyx =µ∂u
∂y
called Newton’s law of viscosity. For materials which do not obey this law, i.e., the shear stress and shear rate are not directly proportional but are related by some function, the fluid is called non-Newtonian [213]. Many of industrial liquids show non-Newtonian behavior, see for example [22],
[27]-[29], [213]. The physical origin of a non-Newtonian behavior relates to the microstructure of the material. Materials such as slurries, pastes, gels, drilling mud, paints, foams, polymer melts or solutions are examples of non-Newtonian fluids. For such fluids the viscosity is not constant, it is a function of either the shear rate or the shear stress.
Many papers have concentrated on the prediction of rheological proper-ties under shear using molecular dynamics computations ([89], [90], [226]).
The observations of these simulations greatly assist us in understanding the behavior of lubricant properties. Various models are employed to represent the viscous or rheological behavior of fluids of practical interest. Frequently, the fluid is treated with the non-Newtonian viscosity function given in terms of the shear rate.
A chief difficulty in the theoretical study of non-Newtonian fluid mechan-ics is to define this relationship. The apparent viscosity µapp is the ratio of shear stress and shear rate, then
τyx =µapp∂u
∂y
holds. We shall investigate boundary layer problems for non-Newtonian fluids whose apparent viscosity depends only on the rate of strain. The actual mathematical form of µapp for these materials will depend on the nature of the particular material. The flow behavior of fluids determined by their rheological properties is described by the relationship between the shear stress and shear rate. This relationship is determined experimentally. The time-independent viscoinelastic fluids are often represented by
(1.9) µapp =K φ
∂u
∂y
, where φ is an empirically determined function.
The most common flow model is the so-called power-law model or the Ostwald-de Waele power-law model, given by [195]. Throughout this work we apply this model when the flow behavior of the non-Newtonian fluid is described by
This provides an adequate representation of many non-Newtonian fluids over the most important range of shear. The shear stress is related to the strain rate ∂u/∂y by the expression
(1.11) τyx=K
whereK andn >0 are a positive constants called consistency and power-law index, respectively, and defined by Bird [26]. The case 0 < n <1 corresponds to pseudoplastic fluids (or shear-thinning fluids), the case n > 1 is known as dilatant or shear-thickening fluids. For n = 1, one recovers a Newtonian fluid. The deviation ofn from a unity indicates the degree of deviation from Newtonian behavior [13].
It should be noted that without the above boundary layer simplifications the dynamic viscosity µ in (1.4) and (1.5) for power-law fluids is calculated by the following relationship
The two-parameter relations (1.11) or (1.12) have been useful in fitting rheological data for a large variety of fluids (see [153], [179]). Parameters K and n are determined empirically. Relation (1.11) may fail to fit the total range of experimental data for some materials. However, the formula can be fitted well to measured data over a restricted range of shear rate. The properties of the material undergoing thermal processing must be known and appropriately modeled to accurately predict the resulting flow and transport, as well as the characteristics of the final product. Some of the values of n are shown in Table 1.1 ([57], [63], [161]). In process industries most non-Newtonian fluids are pseudoplastic (n <1).
The ”functionalization” of solid and fluid materials by addition of chem-ical compound is a process that is going back to Maxwell [147], [148] and Rayleigh [170]. The effective viscosity was characterized by Einstein [80], [81]. Due to measurements for crude oil and experiments with various liq-uids ranging from simple molecular liqliq-uids to polymer melts it was shown that the apparent viscosity dependent on the shear rate. On the base of measurements the authors observed for the apparent viscosity a power law of the form (1.10). The dependence of the powern on many factors (e.g. the pressure, the temperature and the film thickness) was confirmed by simula-tions ([2], [133], [134], [79]). Application of molecular dynamics to rheology has helped to understand the behavior of non-Newtonian fluids to predict quantitative rheological properties such as the viscosity of lubricants [116].
Some lubricants, e.g. silicone fluids and polymer solutions are described by the non-Newtonian power-law model due to the model’s simplicity (see [154], [181], [182], [183], [184]).
Material n
clay suspension 0.1
suspensions (kaolin in water, bentonite in water) ≈0.1-0.15
cosmetic cream ≈0.1-0.4
toothpaste 0.3
molten polymers ≈0.3-0.7
drilling fluid (oil based mud), paper pulp, latex paint ≈0.4-0.6
lubricants ≈0.4-1.1
water, glycerin 1
slurry (sand-water mixture) ≈1.1-1.4
saturated honey ≈1.5
Table 1.1
Thompson et al. [199] showed that at high shear rates the viscosity of glassy films obeys a power law in the form (1.10) with n = 2/3. In [41] and [52] the power exponent n was determined for a polyethylene and n ≈0.3 was obtained in the range of temperature 160◦C-180◦C. It was shown that both K and n depend on the temperature. In case of sand-bentonite-water mixtures for different sand volumetric concentrations the experiments gave n ≈ 1.4 and here the rheological parameters K, n do depend on the volumetric concentration [49].
For simplicity, in our calculations we assume that the power-law exponent n and the consistency K are constants. Even if the main advantage of the power-law model is its relative simplicity, the non-Newtonian behavior of the material complicates the viscous terms in the momentum equation.
Remark. Due to the wide range of applications a large number of ar-ticles has been devoted to different mathematical aspects of the so-called p-Laplacian operator ∆pu =∇(|∇u|p−2∇u). Here, instead n the p parameter is applied. On the mathematical examinations of the solutions to the equa-tions involving ∆p we mention the book by Doˇsl´y and Reh´ak [77], Dr´abek and Milota [78] and also some of my papers [31]-[37]. That type of nonlin-earity appears also in nonlinear diffusion equations arising from a variety of diffusion phenomena ([36], [216]).